PlasmaSources Sci. Technol. 3 (1994) 169-176. Printed in the UK
`
`I Electrical characteristics and electron
`heating mechanism of an inductively
`coupled argon discharge
`
`\' A GG&j&, R 3 p$js!: axd 3 !& A!e=xdr=vich
`Osram Sylvania Inc., 100 Endicoit Street, Danvers. MA 01923, USA
`Received 6 August 1993, in final form 8 October 1993
`
`Abstract. The external electrical characteristics of inductively coupled argon RF
`discharGes at 13.56 MHz have been measured over a wide range of power at gas
`pressures ranging from 3 mTorr to 3 Torr. External parameters, such as coil voltage,
`current and phase shift, were measured. From these measurements the equivalent
`discharge resistance and reactance, the power transfer efficiency and the coupling
`coefficient between the primary coil and the plasma were determined as a function of
`discharge power and gas pressure. The efficient RF power transfer and the large
`value of the effective eiectron collision frequency found here at low gas pressure
`suggest some collisionless electron heating mechanisms. This mechanism is
`identified as non-local electron heating in the inhomogeneous RF field due to spatial
`dispersion of the plasma conductivity.
`
`1. Introduction
`
`Low-pressure inductively coupled RF discharge sources
`have important industrial applications mainly because
`they can provide a high-density electrodeless plasma
`source with low ion energy and low power loss in the
`sheaths. These attractive features of inductively coupled
`discharges are recognized by both the plasma processing
`and the lighting community and therefore the study of
`these types of discharges has been actively pursued,
`especially over recent years.
`Although inductive discharges are being vigorously
`studied [l-41, few data can be found in the literature
`of
`the exiernai &&cai &racte&&
`induc-
`tion coil that drives the discharge. This is not to say that
`voltage and current measurements of the induction coil
`for a specific point cannot be found in the literature but
`that a comprehensive set of measurements over a wide
`range of inductive discharges operation is indeed un-
`available. A comprehensive set of measurements could
`provide a basis for the development of scaling laws for
`the external and internal discharge characteristics from
`which control of RF discharges and optimization of
`piasma processes could be achieved. The measurwieiiiii
`of discharge and plasma parameters over a wide range
`of external conditions given here is valuable for induc-
`tive RF discharge modelling and possibly as a database
`for comparison of theory with experiment.
`The objective of this work is to report on measure-
`ments of the external electrical characteristics of an
`inductively coupled discharge over a wide range of
`operating parameters. Measurements of the voltage,
`current and phase shift on the primary coil of an
`09650252/94/020169 + 08 $19.50 0 1994 iOP Publishing Ltd
`
`inductive discharge have been made over a range of
`power between about 20 W and 180 W and over a range
`of argon gas pressure between 3 Torr, where eiectron-
`neutral (e-n) collisions dominate, and 3 mTorr where
`e-n collisons are considerably reduced. From these
`measurements, the resistive (ohmic) and reactive compo-
`nents of the primary coil impedance, the power transfer
`efficiency, the minimal maintenance power and the effec-
`tive electron collision frequency have been determined
`based on an approach developed by Piejak et a1 [4]. At
`low gas pressure, the effective electron collision fre-
`quency has been shown to exceed the electron-atom
`collision frequency considerably, suggesting that a col-
`iisioniess dissipative process is responsibie For eiectron
`heating. This heating is shown to be the result of spatial
`dispersion of the plasma conductivity in an inhomo-
`geneous RF field, a phenomenon underlying the anomal-
`ous skin effect. The estimation of the effective electron
`collision frequency based on this concept is in reason-
`able agreement with experiment.
`
`2. Experimental set-up
`
`Measurements were made in a discharge chamber for-
`med by a glass cylinder with an ID of 14.3 cm and an OD
`of 15.0 cm and limited at each end of the cylinder by
`aluminium plates 6.7 cm apart. A sketch of the discharge
`vessel and location of the discharge induction (primary)
`coil is shown in figure l(a). All measurements were made
`with argon gas flow to enhance gas purity. The dis-
`charge chamber and vacuum-gas flow system have been
`described elsewhere [SI. This set-up varies from previ-
`
`169
`
`Page 1 of 8
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`Samsung Exhibit 1015
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`

`V A Godyak et al
`
`two
`the
`measurements. The agreement between
`measurement techniques is generally within a few per
`cent. Errors in voltage and current measurements in this
`system c5-n are less than or equal to about 5% while
`error in relative phase shift is about 0.1" which corre-
`sponds to about a 10% error in the worst case of the
`smallest power factor (cos r$ = 0.02) measured in this
`work.
`
`3. Measurement results
`
`yY"'LCLLL""1,
`
`The primary current of the induction coil against total
`power is shown in figure 2. Only data for two gas
`pressures are shown in this figure but these data are
`-F ,4:nnhqvma h*he.r:-..r I + "11
`r r n o l ; + ~ t ; ~ i ~ l r r mnrpron+ot:.ra
`CLL all
`V1 U ' ~ " , , ' u ~ ~
`'L.p'u'""L'L'*'
`"I.II'zL*IV"I
`gas pressure and power levels considered here. At a fixed
`gas pressure the primary current increases with power
`while at a given power level the primary current is.
`smaller at the higher gas pressure. The RF voltage across
`the primary coil against the total power is shown in
`figure 3. The trends in coil voltage against power are
`qualitatively similar to that of the coil current. The
`ous work [4] in that the induction coil is attached to the
`power factor for these two gas pressures is shown in
`exterior of the glass cylinder. The induction coil consists
`figure 4. In all cases the power factor increases more
`of two turns of 0.3 cm wide copper foil strip separated
`lower
`pa.~ei
`pirwei ;eve;s ai,;
`.*$h
`less
`by 2.0 cm and is connected as shown in figure 1. With
`rapidly at higher power levels. At a given power level,
`the coil and aluminium plates positioned as shown and
`the power factor increases with gas pressures.
`no plasma present, the effective inductance Lo and
`The external electrical characteristics of an inductive
`resistance Ro of the unloaded coil (including leads to the
`discharge may be given in terms of the equivalent coil
`matching system) were measured at 13.56 MHz and
`resistance and reactance against total RF power. The
`found to be 0.84pH and 0.65Q respectively. In this
`primary coil resistance against total power is shown in
`paper the inductor with plasma present (loaded) will be
`figure 5. At each gas pressure the equivalent resistance
`referred to as the primary coil, otherwise (with no
`of the primary coil increaser; with power and reaches a
`plasma~~ii Will be ~referred~to~as
`
`t~he~~un~o~ade~~c~oil, Note
`plateau at the higher gas pressures. For a fixed power,
`that the aluminium plates reduce the initial coil induc-
`the primary resistance increases with gas pressure. The
`tance by about 15% suggesting that the RF field is
`increase in equivalent resistance represents coupling of
`localized about the coil winding and loosely coupled to
`the plasma resistance into the primary coil circuit and is
`the plates.
`fundamental in the power transfer from the primary coil
`A simplified schematic diagram of the matcher cir-
`to the secondary (plasma) of an air coil transformer.
`cuit showing the electrical measurement points is given
`Since with increasing power the resistance of the pri-
`in figure l(b). The source of discharge power is an RF
`amplifier that delivers power to a link coupled matching
`system. The secondary coils of the matcher are arranged
`so as to form a symmetric (push-pull with respect to
`I-..--n
`FA.
`---.+-a\
`+ha ;-dl..-+inn +nil nnA
`-C n= --I.._
`~ L V U L , U , D V Y l r r V. nC y'V.YL.1 1".
`L L I L . lll..Ylll"Il U".. Y.I..
`discharge. For a given discharge current, a symmetric
`source reduces the capacitive.coupling which may affect
`the electrical characteristics of the discharge, especially
`at low RF power. The RF voltage is measured with a
`voltage divider directly at the input of the induction coil
`and current is measured with a current transformer as
`shown in figure l(b). These measurements are made with
`a vector voltmeter in which the phase shift between the
`current and voltage is also measured. The incident and
`reflected powers are also measured in
`iine beiween
`the RF source and the matcher. In conjunction with a
`calibration curve to account for matcher losses [SI,
`these measurements serve as an independent check of
`the power determined from the vector voltmeter
`
`0
`P
`
`9
`!!?!?
`!SO
`total RF power (W)
`Figure 2. Current (RMS) through the primary induction coil as
`a function of the total RF power delivered to the coil for gas
`pressures of 0.01 and 1.0 Torr.
`
`200
`
`170
`
`Page 2 of 8
`
`

`

`500 /.
`-
`
`400
`
`2
`
`I
`
`7 5
`
`Characteristics of inductively coupled Ar RF discharges
`
`7
`
`100
`
`0
`a
`
`l o o
`50
`150
`told RF power (W)
`Figure 3. Voltage (RMS) across the Drimaw coil as a function
`ofihe total RF power delivered to the coil fir gas pressures of
`0.01 and 1 .O Torr.
`
`200
`
`0 '
`0
`
`150
`50
`100
`total RF power (W)
`Figure 4. Power factor against total RF power over a power
`range between 20 and 180 W.
`
`I
`200
`
`6
`
`I 4 t
`
`I
`200
`
`0 '
`0
`
`40
`
`60
`120
`160
`total RF power 0
`Figure 5. Primary resistance against total RF power for
`argon gas pressures between 3 miorr and 3 Torr.
`mary coil itself Ro is virtually constant, the increase in
`primary resistance is directly related to an increase in
`the power transferred to the plasma.
`The equivalent primary coil reactance versus power
`is given in figure 6. At a fixed gas pressure the primary
`reactance decreases with total power while at a fixed
`power level the primary reactance decreases with in-
`creasing gas pressure. As in the case of equivalent
`resistance tine reduction in primary reactance refiecis an
`
`0
`
`150
`50
`100
`total RF power (W)
`Figure 6. Primary coil reactance against total RF power for
`argon gas pressures between 3 mTorr and 3 Torr.
`
`200
`
`increase in the mutual coupling between the coil and the
`plasma. Essentially, the plasma current flow partially
`neutralizes the time varying magnetic flux created by
`current flow through the primary coil. Thus, a reduction
`in the primary equivalent reactance represents the dia-
`magnetic effect of the plasma coupled to the primary
`coil.
`
`4. Evaluation of internal discharge parameters
`
`From direct measurements of the external electrical
`characteristics of the discharge, spatially integrated in-
`ternal discharge parameters can be inferred [4]. These
`parameters are the power dissipated in the plasma P,,
`the minimal inductive discharge power Po, the coupling
`coefficient k between the primary coil and the plasma
`and the Q factor of the secondary (plasma) circuit Q,
`(which is the ratio between the reactance and the resis-
`tance of the secondary circuit including the plasma).
`Some of these parameters are unique functions of exter-
`nal (given) discharge parameters: geometry, frequency,
`gas type and pressure and total RF power while others
`may be expressed as an average over a certain RF power
`range or over the range of another external parameter.
`The basis for inferring these internal parameters is the
`transformer theory equations (see, for example the work
`of Piejak et a[ [4]) which in essence are spatial integrals
`of Maxwell's equations:
`p = RI - R, = w2k2LoL,R,/Z4
`(1)
`i = w(Lo - L,) = o Z k Z L o L 2 ( ~ L , + w/veffR,)/Z$
`(2)
`where p and correspond to changes in primary resis-
`tance and reactance (for a series equivalent circuit) due
`to plasma loading; R, and oL, = X , correspond to the
`unloaded coil resistance and reactance; R I and oL,
`correspond to the primary resistance and reactance with
`plasma; L, and R, are the magnetic inductance and
`ohmic resistance (plasma resistance) of the secondary
`circuit; Z , = COL, + O / V , , ~ R , ) ~ + (Rz)2]1'2 is the im-
`pe&nce of ihe nei-ondary &c.uii; and v.ff is ihe e,Teciive
`
`171
`
`Page 3 of 8
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`

`

`V A Godyak et a/
`
`Q2
`
`electron collision frequency accounting for RF power
`dissipation process in the inductive RF plasma [4].
`Expressions for k, Q, and P, directly follow fcom
`equations (1) and (2):
`kZ = (5' + Pz)/Xo(i - P 4 V A
`(3)
`0LzlR2 + &.rr = UP
`(4)
`P, = 12p.
`( 5 )
`From the measured external electrical parameters these
`parameters can be found for each point of discharge
`operation. Moreover, they can be found for any arbi-
`trary inductive RF discharge, without prior knowledge of
`a particular discharge geometry or spatial distribution
`of plasma and the electromagnetic field.
`In our experimental arrangement the aluminium end
`plates affect the unloaded coil inductance to an extent
`comparable to that caused by the plasma. However,
`since the plate conductivity and position referenced to
`the unloaded coil remains unchanged throughout the
`experiment, the influence of the piates Is accounted for
`in the unloaded coil characteristic constants X , and R,
`(measured with the plates and no plasma). Only in a
`strong skin effect regime, when RF field distribution is
`significantly affected by the plasma, might one expect
`some change in X , and R,, however, as will be shown,
`this case is not encountered in these experiments.
`Having inferred RF power absorbed by the plasma
`P,, power transfer efficiency q is simply P 2 / P , . Figure 7
`represents q as a function of the total RF power PI for
`argon pressure between 3 mTorr and 3 Torr. In general,
`q ( P J drops at small and decreasing P , as P , ap-
`proaches the minimal maintenance power Po (dissipated
`in the unloaded coil, Po = l:R, as P, +0) needed to
`induce an RF electric field sufficient to maintain the
`plasma. As shown by Piejak et al
`(1 + Wz/V&)V,Z/kzQoOL2
`(6)
`Po
`where V, = (P,R,)l'Z is the ohmic component of the
`secondary voltage and Q, is the Q factor of the unloaded
`
`primary coil. Po varies with gas pressure, since the
`ohmic component of plasma voltage V2 (governed by
`ionization and energy balance) as well as the factor
`w/v,,, depend on the gas pressure. The minimal mainten-
`ance power Po is the power in the primary coil in the
`limit P, + 0 and can be found from the measurements
`of the primary current I , as Po = lfOR,, where I , , = I ,
`as P, -0.
`In figure 8 the primary current I , is shown as a
`function of the discharge power P, for different argon
`pressures. Exirapoiating i,(P,j io the iimit P, = 0 gives
`an estimate of I , , and P,. From figure 8 for p = 1.0, 0.1,
`0.01 and 0.003 Torr the corresponding values of I , , are
`1.7, 2.5, 4.2, and 5.7 A, resulting in a minimal mainten-
`ance power Po of 1.9, 4.0, 11.5, and 21 W respectively.
`Thus, as shown in [4], power loss in the primary coil
`generally increases with decreasing gas pressure thereby
`reducing power transfer efficiency to the plasma load. As
`one can see in figure 7 the power transfer efficiency,
`grows with RF power, and according to [4] reaches a
`oroau maximum ai the condiiion when ihe secondary
`I._. . J ~. . ~ ~ : - ~
`magnetic reactance equals the plasma impedance:
`oL, = R,(1 + w2/v&)'".
`(7)
`It seems (from figure 7) that such a condition is reached
`for relatively large argon pressure (p = 0.3-3 Torr)
`where collisionally dominated electron heating processes
`occur through electron-atom collisions (vcrr = VJ and
`w'/v:" << 1, where v,,
`is the electron-atom collision
`frequency. At these pressures RF power is eficiently
`transferred to the plasma over a wide range of total RF
`power and q % 0.9. At lower argon pressure (p = 3-30
`mTorr) q is somewhat smaller (q = 0.4-0.8) but does not
`drop as dramatically as expected [4]
`for collisional
`heating and d / v & >> 1.
`Preliminary probe measurements of the electron en-
`ergy distribution function in this discharge at p = 10
`mTorr allowed us to estimate (using the electron-atom
`cross section for argon) the electron-atom collision
`frequency v,, % 2.0 x lo6 s-' suggesting that w/vem x
`
`~~~~~
`
`1 ,
`
`
`
`-
`
`I
`
`0 . . .
`
`p = 0.003 Toon
`
`0 0.01
`* 0.1 0 1
`
`~
`
`n
`
`0
`
`0
`0
`
`100
`5 0
`150
`total RF power (W)
`Figure 7. Power transfer efficiency against total RF power for
`argon gas pressures between 3 mTorr and 3 Torr. The
`symbols for pressure are the same as in the previous
`figures.
`
`200
`
`172
`
`0
`0
`
`160
`120
`80
`40
`discharge power (W)
`Figure 8. The primary c current against discharge power
`for p = 0.003, 0.01, 0.1 and 1 Torr.
`
`200
`
`Page 4 of 8
`
`

`

`40. For this value of w/v,, one can then obtain [4] the
`power transfer efficiency:
`g 4 (1 + 4k-2Q0'~/v,,)-' = 0.25
`(8) :
`where the sign of equality on the left-hand side of
`expression (8) corresponds to the matching condition
`(7).
`Assuming a collisional electron heating process at
`p = 10 mTorr, g should be no more than 0.25, whereas
`experimental~data plotted in figure 7 show much larger
`values of q (up to .rl = 0.8). An even larger difference
`between collisional and experimental values of .rl is
`expected for p = 3 mTorr. Thus, at low pressure there is
`some additional non-collisional dissipation process,
`
`such that veri > v. ..
`
`*
`
`5. Effective electron collision frequency
`
`It appears to be possible to infer the value of verf from
`measured electrical characteristics of the primary coil
`using equation (4), although the parameter wL,/R, in
`this equation remains unknown. This can be done in a
`number of ways. First, the ratio w/verr can be found at
`the specific value of P I corresponding to the matching
`condition (7) when the power transfer efficiency q(P,) is
`maximal:
`
`(10)
`
`4 V d f = ~/2(i/P - do.
`(9)
`Unfortunately, at low pressure, the matching condi-
`tion does not appear to be reached within the RF power
`interval of the present experiment. Another way to find
`verr is a t the condition when wLJR, << w/vefT which
`occurs in the limit of small discharge power P,. Noting
`that R, = V:P;', equation (4) may be rewritten in the
`following form:
`5/P wLzP2lVZZ + wlverr = Qz
`from where in the limit as P , ~ . -+ 0 one obtains:
`(11)
`+.rr = U P .
`Experimental values of Q, = u p are given in figure 9 as
`functions of the discharge power P , for p = 10mTorr
`and 1.0 Torr. Extrapolation of the experimental data of
`( / p for p = 10 mTorr to the point where P , = 0 yields a
`value of w/vetr = 2.2 corresponding to verr = 3.9 x IO'
`s-' which is more than an order of magnitude larger
`than the electron-atom collision frequency veri. The
`corresponding calculation for p = 3 mTorr gives veri z
`2.0 x lo7 s-' which also greatly exceeds ye, at- this
`pressure. As seen in figure 9, at p = 10 mTorr, C/p grows
`linearly with P, inferring, according to equation (lo),
`that the ratio wL,/V,Z is nearly constant. This seems
`reasonable since both L, which is governed by the RF
`field and plasma distributions and V, which is propor-
`tional to the plasma RF field, are nearly independent of
`P, at low gas pressures.
`A different behaviour for tip with P , is seen for
`p = LO Torr, A!though as expected for m/ven << 1,
`
`Characteristics of inductively coupled Ar RF discharges
`
`I
`
`.- -
`b 3 -
`P
`a 6 2 -
`
`w
`0
`m
`
`p = 0.01 Torr
`
`o
`
`n
`
`o
`0
`
`
`0
`
`
`
`c
`
`0
`0
`.a p = 1.0 Torr
`
`0
`
`0
`
`1 -
`
`0
`
`
`O
`
`0
`
`200
`
`80
`120
`160
`40
`discharge power (W)
`Figure 9. The Q factor of the secondary circuit against the
`discharge power for 0.01 and 1 Torr.
`
`the value of wL,/V; does not stay
`( / p + O as P , -0,
`constant when P , grows. This is probably the result of
`discharge constriction due to gas heating at higher
`pressure and may also be due to the skin effect, both
`being more pronounced at higher discharge power. Both
`effects decrease the discharge channel cross section and
`move it closer to the primary coil winding, finally
`resulting in a rising coupling coefficient and discharge
`voltage V,. Visual contraction with growing RF power
`could be seen in our experiments at higher argon pres-
`sure. This phenomena of discharge constriction is well
`known in DC discharges at high current and gas press-
`ure.
`A third way of finding w/verr comes directly from
`equation (3).
`wrvefr = u p - (5 + P2)ik2xop.
`(12)
`Here the coupling coefficient k remains unknown, but
`Since k < 1, one can evaluate the lowest possible value
`of v,lf-,,,in (at k = 1) suggesting:
`veri > vermin = wCt/P -(E2 + P ~ ) / X ~ P I - ' . (13)
`The calculation of v,f.min for p = 10 mTorr over the
`entire interval of discharge power (20 W 4 P , i 100 W)
`gives v,f.min between 3.47 x lo7 s-' and 3.27 x lo' s-I
`both of which are just slightly less than verr = 3.9 x IO'
`s - ' found from using equations (10) and (11).
`The influence of the coupling coefficient k on the
`inferred value of veri is shown in figure 10 where ratios
`of w/v,,, calculated using equation (12) are given as
`function of discharge power for different values assigned
`to k. Two interesting features can be seen in figure 10.
`First, for all reasonable values of k (0.3 4 k 4 l.O), in the
`limit of P , = 0 the calculated values of w/vdr converge
`well to the value of w/verr = 2 2 found earlier. Second, if
`one assumes that the real value of k is that one which
`provides constant values for the inferred value of w/verr
`the result would again be w/verr = 2.2 or veri = 3.9 x lo'
`s- '. Thus, the effective electron collision frequency
`found in different ways gives a value of verf = 4 x 10'
`s-' >> veri z 2 x lo6 s-'. This result suggests some non-
`collisional dissipation process responsible for RF power
`
`1 73
`
`Page 5 of 8
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`

`

`V A Godyak et a/
`
`4
`
`Ar 0.01 Torr
`13.56 MHz
`
`0 k n l . 0
`
`m A '
`
` A AO.6
`
`' 10.5
`
`I
`
`0
`
`0 0.3
`
`I
`
`0
`
`150
`100
`5 0
`discharge power (W)
`Figure 10. The inferred ratio of d v , , against discharge
`power for fixed values of k between 0.3 and 1.
`
`200
`
`transfer to the plasma electrons in the low-pressure
`inductive RF discharge.
`Prior to discussing a possible mechanism of col-
`lisionless power transfer the coupling coefficient k
`should be evaluated. This readily follows from equation
`(3) provided that the ratio o/verr is known. For
`high argon pressure (p > 0.3 Torr) where veri = v,, and
`~ / V , , , << TIP = 1 - 3,
`kZ = (5' + p2)/5X,.
`(14)
`The values of k found using equation (14) for p = 1 Torr
`are shown in figure 11. Results for p = 10 mTorr found
`using equation (3) and o/vcrr = 2.2 inferred for this
`pressure are also shown. Note that the values of k found
`for low pressures from equation (3) are very sensitive to
`the particular value of o/verr. Small deviations in w/verr
`lead to significant dispersion in the values of k. That is
`why at low pressures when o/verr 2 1 it seems more
`reliable to find k as that which provides the minimal
`deviation of o/vcrr with discharge power as shown in
`figure 10. Since in this case k is found as some averaged
`value over the range of discharge power, this 'way of
`inferring k is qp!icab!e when p!asm density a d RF
`field distributions do not change appreciably with dis-
`
`-
`
`p - 0.01 Torr
`
`i
`
`m .-
`
`0 '
`0
`
`80
`160
`120
`40
`discharge power (W)
`Figure 11. Coupling coefficient against power dissipated in
`the discharge for gas pressures of 0.01 and 1.0 Torr.
`
`i
`200
`
`1 74
`
`charge power. This takes place at low gas pressure and
`moderate discharge power, when the skin depth is
`not much smaller than the plasma's characteristic
`dimension [2].
`
`6. Collisionless RF heating in an inductively
`coupled discharge
`
`Collisionless electron heating is now commonly accep-
`ted as a primary mechanism in sustaining low-pressure
`capacitive RF discharges. It occurs at the plasma-sheath
`interface as a result of inelastic electron reflection from
`the oscillating RF electrode sheaths. Contrary to capaci-
`tive RF discharges, the RF current in an inductively
`coupled discharge is closed within the plasma and does
`not form an oscillating RF sheath. Since some capacitive
`coupling between the primary coil and the inductive
`plasma (through the glass wall) takes place, one might
`suggest that capacitive coupling may be responsible for
`the collisionless (stochastic) electron heating in the wall
`sheath. However, it has been demonstrated that capaci-
`tively coupled RF discharges maintained in the same
`discharge chamber [7] (using parallel plates as RF elec-
`trodes having an order of magnitude larger surface area
`than the primary coil) dissipate an order of magnitude
`less power than in a inductive discharge under the same
`RF voltage and gas pressure. Thus, capacitive effects on
`the inductive discharge appear to affect the energy
`balance by only about 1% at low pressure. A more
`precise estimation of capacitive effects has been done by
`comparing RF power consumed in the capacitive mode
`with the primary coil cut in the middle with that
`consumed in the inductive mode with an equal RF
`voltage applied to the primary coil. At p = 10 mTorr
`and P, between 20 and 100 W this experiment showed
`that the influence of capacitive effects on the discharge
`energy balance is less than 1%. Therefore, we have to
`conclude that in this. experiment stochastic electron
`heating in the wall sheath is negligible.
`We believe that the collisionless RF power dissipation
`mechanism in low pressure inductive RF discharges (first
`found and reported by Godyak and Piejak [SI) is
`related to the phenomenon, discovered for superconduc-
`tors by Pippard [9] almost a half century ago, known
`as the anomalous skin effect [lo, 111. This effect occurs
`in superconductors and collisionless plasmas when the
`skin depth 6 is smaller than the electron mean free path
`for momentum transfer A,. Under this condition the well
`known formula for normal (resistive) skin depth 6,:
`... m..
`P - P - ..m
`" = om = L,L/LW",
`- ' / W e ( L Y s n / " ,
`I,.. r r p
`(I51
`,-ll2
`= . . I
`is not applicable. Here cr is the real part of the plasma
`conductivity, o, is the electron plasma frequency and c
`is the speed of light in vacuum. Note that equation (15)
`is only valid for v,, >>U. The validity of equation (15)
`for normal skin effect suggests local coupling between
`the RF field and the RF current within the skin layer, i.e.
`d,<<6. For the opposite case when A,
`there no
`longer is local coupling between the RF field and the RF
`
`Page 6 of 8
`
`

`

`,(16)
`
`current within the skin layer. Due to thermal electron
`motion, electrons accelerated in the skin layer create an
`RF current outside of the skin layer. This effect, known
`as spatial dispersion of the plasma conductivity [lo],
`leads to the expression of skin depth for the anomalous
`skin effect [9, 111:
`6 = 6, = c/o,(4v,,o,/cw)"3
`is the electron thermal velocity, v,, =
`where v,.
`(8kT,/mji'2.
`Expression (16) is valid for w << vTe!S, meaning that
`electrons cross the skin layer in less time than the RF
`field period [ll], i.e. the electrons practically move as in
`a DC field, gaining energy from the RF field localized in
`the RF skin layer. This essentially differs from collision-
`less electron motion in a homogeneous RF field where
`electrons gain energy in one half cycle of the RF field and
`return the energy back in the other half of the cycle.
`There is no electron heating in the RF field unless some
`phase mixing mechanism breaks the regularity in the
`electron oscillation. For collisional heating this occurs
`due to electron-atom (and/or ion) collisions. In the case
`of the anomalous skin effect the mixing mechanism is
`provided by the electron thermal motion (spatial disper-
`sion) which moves electrons out of the skin layer to the
`neighbouring plasma with no RF field, thus preventing
`electrons from returning the energy acquired in the skin
`layer back to the RF field.
`The anomalous skin effect outlined here could ex-
`plain the collisionless electron heating in the skin effect
`controlled (skinned) inductive RF discharge, which
`occurs when the RF field space distribution in the plasma
`E(r) is significantly affected by the plasma conductivity
`and differs from that with no plasma Eo(r), but we have
`no evidence from our experiment that this is the case. '
`Our estimation of the skin depth for p = 10 mTorr and
`P, = 50 W using formulae (15) and (16) and that for the
`non-dissipative (reactive) skin depth So = c/o,, all give
`about the same value far 6, z 6- z 6, z 3-3.5 cm.
`Moreover, these are very close to the characteristic
`length of the RF field inhomogeneity in the absence of a
`= 3 an. For the coil
`plasma, A, = [grad In
`configuration in our experiment, A, is determined by the
`coil pitch rather than by its radius. The estimated
`numbers cannot support either a certain kind of skin
`effect or even the importance of the skin effect at all
`under the conditions of our experiment. However, we
`would like to state here that the space dispersion
`.. ..._.. governs !he anomalous skin effect
`mechnniem "... which
`should work in a low-pressure inductive discharge inde-
`pendent of whether the discharge is in a skinned regime
`or not.
`Indeed, the dissipative mechanism underlying the
`anomalous skin effect originates not by the skin effect
`itself (due to electromagnetic induction caused by the
`plasma current) but by the RF field inhomogeneity which
`is always the case in an inductive RF discharge. In this
`regard there is an essential difference between a long
`homogeneous conductor connected to an RF current
`
`Characteristics of inductively coupled Ar RF discharges
`
`source and a plasma maintained by an RF current
`flowing in the primary coil outside the plasma. In the
`former case the RP field inhomogeneity and accompany-
`ing collisionless dissipation arises as the result of the
`skin effect while in the latter case the inhomogeneous RF
`field and collisionless dissipation occurs whether the
`skin effect is present or not. In other words, in an
`inductively coupled plasma (independent of the plasma
`density) the RF field is inhomogeneous as if there is a
`skin effect!
`Let us now evaluate the effective electron collision
`frequency verr corresponding to the RF energy dissipation
`due to spatial dispersion in the inhomogeneous RF field.
`As pointed out by Pippard [9] for the anomalous skin
`effect, the length of the electron-RF field interaction is
`equal to the skin depth 6, >>A,, contrary to the normal
`skin effect where the interaction length is equal to the
`electron mean free path 1, << 6,. Accordingly, collision-
`less, dissipation due to spatial dispersion in general is
`eoverned by the length of the electron-RF field interac-
`"~
`tion equal to the RF field inhomogeneity length:
`A = b r a d In E(r)]-'
`(17)
`which in the specific case of the skinned discharge
`(6, << Ao) is equal to the skin depth (A = 6J. Therefore,
`vCrf in the expression for the real part of the plasma
`conductivity, U = 0~/4zverr, used in the model [4] and
`in the derivation of the present work, can be evaluated
`as follows
`
`verr = avTe/A
`(18)
`where a is some constant number of the order of one.
`This constant can be found by substituting equation (18)
`in equation (15) and equating it to
`instead of v,,
`equation (16) for the anomalous skin depth. Such a
`procedure yields U = 2 and
`V d f = 2VT,IA.
`(19)
`For A, estimated earlier to be around 3 cm and
`z 8 x !e7 cm s-' (ca!cn!atd from probe mcernre-
`ments) found from equation (19) vel? = 5.3 x lo' s-'
`is very close to that from our experiment, vert=
`3.9 x 107 S-1.
`
`7. Conclusions
`
`Electrical characteristics of the primary coil of an induc-
`tively driven low-pressure RF discharge have been mea-
`sured over a wide range of discharge conditions. These
`measurements show general trends and relationships
`(experimental scaling laws) for basic .electrical para-
`meters of inductively driven plasma as a function of RF
`power and gas pressure. Based on a macroscopic analy-
`sis approach previously developed [4] some discharge
`integral characteristics, important for practical use of
`inductively driven plasmas, have been inferred. Because
`of the complicated geometry in this experiment it is
`non-trivial to infer values of the RF field and plasma
`density as had been done in a collision dominated
`
`175
`
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`

`V A Godyak et a/
`
`inductive discharge with a long cylindrical coil [4].
`Nonetheless, this can be done by making some assump-
`tions (or numerical calculations) with r